Robert Wisbauer
University of D¨ usseldorf, Germany e-mail: wisbauer@math.uni-duesseldorf.de
April 10, 2007
Abstract
Algebras and coalgebras are fundamental notions for large parts of mathematics. The basic constructions from universal algebra are now expressed in the language of categories and thus are accessible to classical algebraists and topologists as well as to logicians and com- puter scientists. Some of them have developed specialised parts of the theory and often reinvented constructions already known in a neigh- bouring area. One purpose of this survey is to show the connection between results from different fields and to trace a number of them back to some fundamental papers in category theory from the early 70’s.
Another intention is to look at the interplay between algebraic and coalgebraic notions. Hopf algebras are one of the most interesting objects in this setting. While knowledge of algebras and coalgebras are folklore in general category theory, the notion of Hopf algebras is usually only considered for monoidal categories. In the course of the text we do suggest how to overcome this defect by defining a Hopf monad on an arbitrary category as a monad and comonad satisfying some compatibility conditions and inducing an equivalence between the base category and the category of the associated bimodules. For a set G, the endofunctor G× − on the category of sets shares these properties if and only ifG admits a group structure.
Finally, we report about the possibility of subsuming algebras and coalgebras in the notion of (F, G)-dimodulesassociated to two functors F, G : A → B between different categories. This observation, due to Tatsuya Hagino, was an outcome from the theory of categorical data types and may also be of use in classical algebra.
Contents: 1.Introduction, 2.Modules and comodules, 3.Relations be- tween functors, 4.Relations between endofunctors, 5.Combining monads and comonads, 6.(F, G)-dimodules.
1
1 Introduction
The purpose ofUniversal Algebrais to develop a general theory which applies to a large part of algebraic structures. Initially it started with the study of abstract algebras, that is, sets A with a collection of (n-ary) operations on A. In his thesis (1963, [32]) Lawvere suggested to formulate these general settings in the language of categories and functors.
In particular it turned out that adjoint pairs of functors are of central importance. The study of monads (triples) and their modules and comonads (cotriples) and their comodules was initiated by Eilenberg and Moore in [21]
and in Beck [5]. This approach covers an extremely wide range of applications including not only the classical notions from module theory and topology but also those from model theory and logic, the latter being of growing interest for computer science.
Bringing the notions down to the language of categories and functors, the common part of all theories envisaged has to be expressed in this terminology.
Many interesting observations in these areas can be traced back and under- stood with the abstract handling of functors as suggested by Eilenberg, Mac Lane and their schools. To make this clear we will pay some attention to the so calledmixed distributive lawsbetween endofunctors introduced in Beck [6]
and further studied in van Osdol [54] and Wolff [57] (see also Barr and Wells [4] and ˇSkoda [46, Section 9]). These were rediscovered in Turi and Plotkin [52] in the context of operational semantics and by Brzezi´nski and Majid [14]
in the form ofentwining structuresbetween the tensor product of an algebra and a coalgebra over a commutative ringR. Some of these notions were more generally handled in monoidal categories by Mesablishvili in [40]. The action of monoidal categories on any category as considered by ˇSkoda in [45] puts this in a wider context.
We note that for some purposes it is natural to consider monads and comonads in 2-categories and we refer to Street [47], Lack and Street [31], Power and Watanabe [43], Lenisa, Power and Watanabe [33] and Tanaka and Power [51] for a treatment in this direction. Szlach´anyi also considers 2-categories in [48] to understand (op)monoidal functors for bialgebroids. His approach is similar in spirit to our investigations but his focus is on different results.
Now each field of application has its own requirements and the questions and constructions arising, say, in computer science may also be of relevance for classical modules. In fact this lecture was motivated by the observation that manipulations suggested and performed by computer scientists (e.g. [52]) are also of interest for corings and comodules.
Dualising the algebraic theory yields the coalgebraic theory and both of these have their realms of applications. However, there are also combinations
of constructions from both of them and one purpose of this presentation is to outline how this can be done and what the outcome is.
As some other authors do (e.g. [11], [45]) we modify the terminology from Eilenberg and Moore in [21] by referring to theF-algebrasof an endofunctor in their sense as F-modules, and F-coalgebras as F-comodules. This is in total correspondence to the application in module theory where the functor F plays the role of an algebra or coalgebra, respectively, while the attached categories are modules or comodules. We hope this change does not lead to conflicts in other situations.
At the core of our treatise is Johnstone’s general Lifting of functors the- orem 3.1 from which many applications can be understood and derived. A different handling of related problems can be found in Mesablishvili [39] where a generalisation of descent theory is proposed.
As a byproduct, on our way we obtain some of the conditions on a monad S on a monoidal category C from Moerdijk [41, Proposition 1.4] which allow the tensor product to lift from C to the category of S-modules (see 3.4).
Monads of this type were named Hopf monads in [41], called opmonoidal monad in McCrudden [37, Example 2.5] and Szachl´anyi [48, Definition 2.4], and bimonads in Brugui`eres and Virelizier [11, 2.3]. We do not use these notions but will associate a different meaning to the termHopf monad here.
Classical Hopf algebras H over a commutative ring R have compatible algebra and coalgebra structures H⊗RH →H and H →H⊗RH such that the free functorφH :R-Mod→R-ModHH induces a category equivalence (e.g.
[16, 15.5]). Traditionally most generalisations of Hopf algebras are based on a kind of tensor product on a category which allows us to follow the usual arguments at least for a larger part. Trying to have the aforementioned equiv- alence in utmost generality for an endofunctorB on an arbitrary category A, one has to impose all conditions necessary on the functor B. Thus we re- quire B to be a monad and a comonad whose compatibility is controlled by a natural transformation λ : BB → BB (mixed distributive law). We call this a (mixed) bimonad and suggest naming it a Hopf monad provided the related free functor φBB : A→ ABB induces an equivalence (see 5.15). In case λ is derived from a natural transformation τ : BB → BB which satisfies the Yang-Baxter equation, following Takeuchi,B may be called a braided bi- monadorbraided Hopf monadif B is a Hopf monad (see 5.17). We note that natural transformations satisfying the Yang-Baxter equation are also used by Menini and Stefan in [38] to define compatible flip morphisms for monads on arbitrary categories and their concept was extended by Kasangian, Lack and Vitale in [29] (see 5.18). By a suggestion of Manin, in noncommutative geometry the role of the twist map can be replaced by an arbitrary Yang- Baxter operator (on vector space categories). This approach leads to similar formulas and is developed by Baez in [2]. Relations between Yang-Baxter op-
erators and entwining structures (on vector space categories) are investigated by Brzezi´nski and Nichita in [15].
As a special case we can look at a setGand the endofunctorG× −on the category of sets. As outlined in Caenepeel and Lombaerde [19], the existence of an antipode for the related bimonad is then equivalent toGhaving a group structure and implies thefundamental theorem. The notions introduced here deepens the understanding of this situation and in this case the fundamental theorem implies in turn the existence of an antipode (see 5.19, 5.20).
Inspired by questions arising in computer science, Hagino introduced in [25] for two functors F, G : A → B between arbitrary categories the notion of (F, G)-algebras(we call them (F, G)-dimodules). These generalise modules as well as comodules of endofunctors and in Section 6 we give a short outline how to treat them in abstract category theory. Their use in (classical) algebra still awaits exploration. For a deeper presentation of examples and usage of (co)algebras in universal algebra and computer science the interested reader may consult Gumm’s articles [23, 24].
To avoid confusion, we mention that, as usual, functor symbols are writ- ten on the left side of an object. When referring to examples in module categories over a commutative ringR, following the usage of our main refer- ences, we usually consider right tensor functors− ⊗X,X anR-module. The reader should be aware that this causes a reversion of symbols in some of the diagrams or formulas when passing from arbitrary categories to R-modules.
2 Modules and comodules
To fix notation let us recall some basic facts.
2.1. Categories. A category A consists of a class of objects Obj(A) and a class of morphism sets Mor(A) satisfying
(i) for objects A, B inA there is a morphism set MorA(A, B) such that MorA(A, B)∩MorA(A0, B0) = ∅ for (A, B)6= (A0, B0);
(ii) for A, B, C ∈Obj(A) there is a composition map
MorA(A, B)×MorA(B, C)→MorA(A, C), (f, g)7→gf;
(iii) for every A∈Obj(A) there is an identity morphism IA ∈MorA(A, A).
The connection between two categories is given by
2.2. Functors. A covariant functor F : A → B between two categories consists of assignments
Obj(A)→Obj(B), A 7→ F(A),
Mor(A)→Mor(B), f :A →B 7→ F(f) :F(A)→F(B),
such thatF(IA) = IF(A) and F(f g) = F(f)F(g).
Contravariant functors reverse the composition of morphisms. Here all our functors considered will be covariant.
The relation between two functors is described by
2.3. Natural transformations. LetF, F0 :A→B be covariant functors.
A natural transformation α:F →F0 is given by morphisms αA:F(A)→F0(A) in B,A∈Obj(A),
such thatf :A→B inA induces the commutative diagram in B F(A) F(f) //
αA
F(B)
αB
F0(A) F
0(f)//F0(B).
Given another pair of functorsG, G0 :B→Cwith natural transformation β :G→G0, the diagram
GF Gα //
βF
GF0
βF0
G0F G
0α //G0F0
is commutative and thus there is a natural transformation (Godement product) βα :=βF0◦Gα=G0α◦βF :GF →G0F0.
If A is a small category then the endofunctors (as objects) together with natural transformations form a strict monoidal category: the product of end- ofunctors is the composition and the composition of natural transformations is given by the Godement product (e.g. [46, Section 8]).
In what follows we will use functors and natural transformations as basic tools for general constructions.
2.4. Adjoint pairs of functors. A pair (L, R) of functors L : A→ B and R : B→ A between categories A and B is called adjoint if there are natural isomorphisms (in A∈Obj(A) and B ∈Obj(B))
ϑA,B : MorB(L(A), B)→MorA(A, R(B)).
Associated to such a pair there are natural transformations unit η:IA→RL and counit ε:LR→IB.
2.5. F-modules. Given an endofunctorF :A→A, an F-module (A, %A) is anA∈Obj(A) with a morphism in A,
%A:F(A)→A.
A morphism f : A → A0 in A between F-modules is an F-module mor- phismprovided it induces a commutative diagram
F(A) F(f)//
%A
F(A0)
%A0
A f //A0.
With these morphisms, the F-modules form a category which is denoted byAF. There is the faithful forgetful functor
UF :AF →A, (A, %A)7→A.
In all generality some limits in AF are induced from limits in A.
2.6. Proposition. Let L:Λ→AF be any functor and Λ a small category.
(1) If lim←−L exists in A, then it belongs to AF.
(2) Assume that F preserves colimits. If lim−→L exists in A, then it belongs to AF.
The relations betweenAF andAare even stronger if additional conditions are imposed on the endofunctor F.
2.7. Monads. A monadon Ais a triple F= (F, µ, η), where F :A→Ais a functor and
µ:F F →F, η:IA→F, are natural transformations with commutative diagrams
F F F µF //
F µ
F F
µ
F F µ // F ,
F ηF //
F η
=
""
FF FF FF FF
F F F
µ
F F µ // F .
Given two monads F = (F, µ, η) and F0 = (F0, µ0, η0) on A, a natural transformation α : F → F0 is called a morphism of monads if the following induced diagrams commute:
F F αα //
µ
F0F0
µ0
F α //F0,
IA η //
ηB0BBBBB
BB F
α
F0.
2.8. Monads and their modules. Given a monadF= (F, µ, η) onA, anF- module is anA∈Obj(A) and a morphism%A:F(A)→A with commutative diagrams
F F(A) µA //
F %A
F(A)
%A
A ηA //
ICACCCCCC!!
CC F(A)
%A
F(A) %
A
// A , A .
As shown in Eilenberg-Moore [21], for a monad F, the forgetful functor UF :AF →A is right adjoint to the (free) functor
φF :A→AF, A 7→ (F(A), F F(A)−→µA F(A)), A→f A0 7→ F(A)−→F(f) F(A0),
by the isomorphisms for A∈Obj(A) and B ∈Obj(AF),
MorAF(F(A), B)→MorA(A, UF(B)), f 7→f◦ηA. Notice that UFφF =F.
Dual to the preceding notions there is a theory of comodules which we sketch in the next paragraphs.
2.9. G-comodules. For a functor G : A → A, a G-comodule (A, %A) is an A∈Obj(A) with a morphism in A,
%A:A→G(A).
A morphism f : A → A0 in A between G-comodules is a G-comodule morphisminducing the commutative diagram
A f //
%A
A0
%A0
G(A) G(f)//G(A0).
The G-comodules together with G-comodule morphisms form a category which we denote by AG. The forgetful functor is faithful,
UG :AG→A, (A, %A)7→A.
As a sample of the relation between the categories Aand AGwe mention:
2.10. Proposition. Let L:Λ→AG be any functor and Λ a small category.
(1) If lim−→L exists in A, then it belongs to AG.
(2) Assume that G preserves limits. If lim←−L exists in A, then it belongs to AG.
For more information about the behaviour of limits and colimits and ex- amples of modules and comodules in the category of sets we refer to Ad´amek and Porst [1].
2.11. Comonads. A comonad is a triple G = (G, δ, ε), where G:A→ A is a functor and
δ :G→GG, ε:G→IA, are natural transformations with commuting diagrams
G δ //
δ
GG
Gδ
GG δG // GGG ,
G δ //
δ
=
""
FF FF FF FF
F GG
εG
GG Gε //G.
Given two comonads G = (G, δ, ε) and G0 = (G0, δ0, ε0), a natural trans- formation β : G → G0 is called a morphism of comonads if the following diagrams commute:
G
δ
β //G0
δ0
GG ββ //G0G0,
IA ε //
β
G0 G0.
ε0
==|
||
||
||
|
2.12. Comonads and their comodules. LetG = (G, δ, ε) be a comonad.
A G-comoduleis an object A∈Obj(A) with a morphism
%A:A→G(A) in A inducing commutative diagrams
A %
A //
%A
G(A)
δA
A %
A //
IGAGGGGGG##
GG
G G(A)
εA
G(A) G%
A //GG(A), A.
The forgetful functor UG :AG→A is left adjoint to the (free) functor φG:A→AG, A 7→ (G(A), G(A)−→δA GG(A)),
A→f A0 7→ G(A)−→G(f)G(A0),
by the isomorphisms
MorAG(B, G(A))→MorA(UG(B), A), f 7→εA◦f, for any A∈Obj(A) and B ∈Obj(AG). Notice thatUGφG =G.
We now recall that monads and comonads are closely related to adjoint pairs of functors (e.g. [21]):
2.13. Adjoint pairs and (co)monads. Let L:A →B and R : B→A be an adjoint pair of functors (see 2.4) with
unit η:IA→RL and counit ε:LR→IB, Then RL:A→A has a monad structure with
productµ=RεL :RLRL→RL and unit η:IA→RL, and LR:B→B has a comonad structure with
coproductδ =LηR:LR →LRLR and counitε:LR→IB.
3 Relations between functors
To study the relationship between various module categories, the following definition is of interest. It was formulated in Johnstone [27] for monads but we also consider it for arbitrary endofunctors.
3.1. Lifting of functors. LetF and G be endofunctors of the categories A and B respectively. Given functors
T :A→B, T :AF →BG, and Tb:AF →BG
we say thatT (resp.Tb)is a lifting ofT provided the left (resp. right) diagram AF
T //
UF
BG UG
A T //B,
AF
Tb //
UF
BG
UG
A T //B, is commutative, where theU’s denote the forgetful functors.
The following assertions are easy to verify.
3.2. Proposition. With the notation in 3.1, consider the functors T F, GT :A→B.
(1) For any natural transformation λ:GT →T F, the functor T :AF →BG, (A, %A) 7→ (T(A), T(%A)◦λA),
A−→f A0 7→ T(A)−→T(f) T(A0), is a lifting of T :A→B.
(2) For any natural transformation ϕ:T F →GT, the functor Tb:AF →BG, (A, %A) 7→ (T(A), ϕA◦T(%A)),
A−→f A0 7→ T(A)−→T(f) T(A0), is a lifting of T :A→B.
For monads and comonads there exist bijections between liftings and cer- tain natural transformations.
3.3. Lifting for monads. (Applegate) Let F= (F, µ, η) and G= (G, µ0, η0) be monads on the categories A and B, respectively, and let T : A → B be a functor.
(1) The liftings T :AF → BG of T are in bijective correspondence with the natural transformations λ:GT →T F inducing commutative diagrams
GGT µ
0
T //
Gλ
GT
λ
GT F λF //T F F T µ //T F,
T η
0 T //
T ηCCCCCCC!!
C GT
λ
T F.
(2) If AF has coequalisers of reflexive pairs and T has a left adjoint, then any lifting T has a left adjoint.
(3) Assume there is a liftingT :AF →BG with invertible λ:GT →T F. If T has a right adjoint, then T has a right adjoint.
Proof. This is proved in Johnstone [27, Lemma 1, Theorem 4 and 2].
For λ, the left hand diagram corresponds to λA being a G-module mor- phism for every F-module (A, %A) and the right hand diagram is related to unitality.
Given a lifting T : AF →BG, λ is obtained in the following way. For any A∈Obj(A) there is an isomorphism
αA : MorBG(φGT(A), T φF(A))→MorB(T(A), UGT φF(A)).
Then T(ηA) : T(A) → T F(A) = T UFφF(A) = UGT φF(A) belongs to the right side of the isomorphism and we obtain ¯λ(A) =α−1A (T(ηA)) :φGT(A)→ T φF(A). This yields
λ=UGλ¯:GT =UGφGT →UGT φF =T UFφF =T F.
t u
3.4. Lifting of a tensor product to modules. The above proposition can be applied to characterise monads (S, µ, η) on a monoidal category (C,⊗, E) (see [35]) for which the monoidal structure from C lifts to the category CS of S-modules. The situation is described by the diagram
CS × CS
−⊗−//
US×US
CS US
C × C −⊗− //C,
By 3.3(1), for the existence of some−⊗− making the diagram commute one needs a natural transformation
λ :S(X⊗Y)→S(X)⊗S(Y) yielding commutative diagrams
SS(X⊗Y) µX⊗Y //
Sλ
S(X⊗Y)
λ
S(S(X)⊗S(Y)) λS //SS(X)⊗SS(Y)µX⊗µY//S(X)⊗S(Y), X⊗Y ηX⊗Y//
ηOXOO⊗ηOOOYOOOOO''
O S(X⊗Y)
λ
S(X)⊗S(Y).
This corresponds to the first three diagrams in [41, Section 7].
To make CS a unital tensor category one has to require that E is an S- module for some morphismS(E)→E inC and that the coherence conditions are respected (diagrams (4) to (7) in [41, Section 7]). Such monads are called Hopf monads in Moerdijk [41] (see introduction).
3.5. Lifting for comonads. LetF= (F, δ, ε)andG= (G, δ0, ε0)be comonads on the categories A and B, respectively, and let T :A→B be a functor.
(1) The liftings Tb:AF → BG of T are in bijective correspondence with the natural transformations ϕ : T F → GT inducing commutativity of the diagrams
T F T δ //
ϕ
T F F ϕF //GT F
Gϕ
GT δ
0
T //GGT,
T F T ε //
ϕ
T GT.
ε0T
=={
{{ {{ {{ {
(2) If AF has equalisers of reflexive pairs and T has a right adjoint, then any lifting Tb has a right adjoint.
(3) Assume there is a lifting Tb : AF → BG with ϕ : T F → GT invertible.
If T has a left adjoint, then the lifting Tb has a left adjoint.
Proof. Dual to 3.3. tu
3.6. Lifting of a tensor product to comodules. Let (T, δ, ε) be a comonad on a monoidal category (C,⊗, E) for which the monoidal structure fromC lifts to the categoryCT ofT-comodules. The situation is described by the diagram
CT × CT −⊗−b //
UT×UT
CT
UT
C × C −⊗− //C,
By 3.5(1) for the existence of some−⊗−b making the diagram commute one needs a natural transformation
ϕ :T(X)⊗T(Y)→T(X⊗Y)
and a morphism E → T(E) yielding the commutative diagrams as required by 3.5 plus appropriate conditions to assure the coherence conditions.
4 Relations between endofunctors
In this section we will specialise the preceding observations to A = B and endofunctors.
4.1. Lifting of the identity. Let F = (F, µ, η), F0 = (F0, µ0, η0) be monads and G = (G, δ, ε), G0 = (G0, δ0, ε0) be comonads on the category A. Then I :AF →AF0 orIb:AG →AG
0 are liftings of the identity if the corresponding diagrams commute:
AF I //
UF
AF0 UF0
A I //A,
AG
Ib //
UG
AG
0
UG0
A I //A.
(1) There is a bijection between the liftingsI of the identity functor and the monad morphisms α:F0 →F.
(2) There is a bijection between the liftingsIbof the identity functor and the comonad morphisms α:G→G0.
Proof. The assertions follow from 3.3 and 3.5. A proof is also given in
Borceux [9, Proposition 4.5.9]. tu
The next observation is a special instance of the situation described above and is the essence of the theory of Galois comodules (see 5.9).
4.2. Adjoint functors to comodules. Let G= (G, δ, ε) be a comonad on the category A and assume there is an adjoint pair of functors with counit
L:B→AG, R:AG →B, ν :LR→IAG, for some categoryB. Then the functors
Rb =RφG :A→B, Lb=UGL:B→A form an adjoint pair, we have the commutative diagram
A
φG //
Rb
?
??
??
??
?oo UG AG
R
~~}}}}}}}}}}}}}}}}
B
Lb
__??
??
??
??
L
>>
}} }} }} }} }} }} }} }}
and for any A∈Obj(A) andB ∈Obj(B) the isomorphisms
MorA(UGL(B), A)'MorGA(L(B), φG(A))'MorB(B, RφG(A)).
Thus the composition LbRb = UGLRφG : A → A is a comonad on A and there is a functorial morphism
νG(A) :LbR(A) =b LRG(A)→G(A), yielding the commutative diagram
ALbRb
Ib //
ULbRb
AG
UG
A I //A,
whereIbis induced by the comonad morphism νG :LbRb→G.
In what follows we will consider the lifting of endofunctors to the category of some modules or comodules.
4.3. Lifting of endofunctors. Let F, G and T be endofunctors of the category A. For the functors T : AF → AF and Tb : AG → AG we have the diagrams
AF T //
UF
AF
UF
A T //A,
AG
Tb //
UG
AG
UG
A T //A,
and we say thatT orTbare liftings ofT provided the corresponding diagrams are commutative.
Besides the situations considered before we may now also ask when the liftings of a monad T are again monads.
4.4. Lifting of monads to monads. Let F = (F, µ, η) be a monad and T :A→A any functor on the category A.
(1) The liftings T :AF →AF of T are in bijective correspondence with the natural transformations λ : F T → T F inducing commutativity of the diagrams
F F T µT //
F λ
F T
λ
F T F λF //T F F T µ //T F,
T ηT //
T ηCCCCCCC!!
C F T
λ
T F.
(2) If T = (T, µ0, η0) is a monad, then the lifting T : AF → AF of T with natural transformationλ:F T →T F is a monad if and only if we have the commutative diagrams
F T T F µ
0 //
λT
F T
λ
T F T T λ //T T F µ
0
F //T F,
F F η
0 //
ηD0FDDDDDD!!
D F T
λ
T F.
(3) For a monad T = (T, µ0, η0), a natural transformation λ : F T → T F induces a canonical monad structure on T F if and only if the diagrams in (1) and (2) are commutative.
Proof. See Beck [6]. To give an idea of the techniques involved we outline some of the arguments.
The assertion in (1) follows immediately from 3.3(1). It is related to the condition onλA being anF-module morphism for any A∈Obj(A).
The diagram in (2) is derived from the requirement that µ0F(A) and η0F(A) are to be F-module morphisms for any A ∈ Obj(A). The first of these
conditions corresponds to commutativity of the right hand rectangle in the diagram
F T T F T T η//
F µ0
F T T F λT F //
F µ0F
T F T F T λF //T T F F T T µ //T T F
µ0F
F T F T η //F T F λF //T F F T µ //T F.
The bottom line is part of the commutative diagram F T F T η//
λ
F T F
λF
T F T F η//T F F T µ //T F
showing thatT µ◦λF◦F T η =λ, while the top line is part of the commutative diagram
F T T λT //
F T T η
T F T
T F T η
T λ
**U
UU UU UU UU UU UU UU UU UU U
F T T Fλ
T F
//T F T F T λ
F
//T T F F T T µ //T T F.
This yields the diagram given in (2).
(3) This is shown in Beck [6] who called the diagrams a distributive law of a monad T over a monad F. (See also [4].) tu Obviously T F being a monad need not imply that T and F both are monads.
An explicit presentation of the material on lifting of monads and endo- functors can be found in Tanaka’s thesis [50].
4.5. Definition. Given two monads F = (F, µ, η) and T = (T, µ0, η0) on a category A, a natural transformation λ : F T → T F is said to be monad distributive provided the diagrams in 4.4(1) and (2) are commutative.
4.6. Tensor product of algebras. Given two R-algebras A, B, and an R-linear map
λ:B⊗RA→A⊗RB,
the tensor product A⊗RB can be made into an algebra by putting (a⊗b)·(a0⊗b0) = a λ(b⊗a0)b0, fora, a0 ∈A, b, b0 ∈B.
IfA and B are associative, the functors − ⊗RAand − ⊗RB are monads on the category ofR-modules. Then the product defined onA⊗RB is associative and unital if and only if − ⊗RA⊗RB is a monad for the R-modules, that is, λ has to induce commutativity of the corresponding diagrams in 4.4. For this special case the conditions are formulated in Caenepeel, Ion, Militaru and Zhu [18, Theorem 2.5].
4.7. Braidings in R-Mod. A prebraiding on the category R-Mod is given by natural morphisms
τB,A :B⊗RA→A⊗RB for any R-modules A, B, satisfying τA,R=τR,A=IA and
τB⊗A,C = (τB,C ⊗IA)(IB⊗τA,C), τB,A⊗C = (IA⊗τB,C)(τB,A⊗IC).
A prebraiding is called a braiding provided all τB,A are isomorphisms. If τA,B ◦τB,A = I for all A, B, then the braiding τ is said to be a symmetry.
Clearly the twist map tw :B⊗RA→A⊗RB, b⊗a7→a⊗b, is a symmetry.
By naturality of τ, for any linear map µ : B ⊗R B → B, we have the commutative diagram
B⊗B⊗A
I⊗τB,A
µ⊗I //B⊗A
τB,A
B⊗A⊗BτB,A⊗I//A⊗B ⊗B I⊗µ//A⊗B.
Similarly, naturality ofτ induces the commutativity of all diagrams appearing in 4.4. Thus for any prebraidingτ, the natural transformation
− ⊗RτB,A : − ⊗RB⊗RA → − ⊗RA⊗RB is monad distributive for all pairs of R-algebras A, B.
Similar constructions can be considered in monoidal categories (e.g. [28], [36], [44]).
In 4.4(2), conditions are given for the lifting of a monad to be a monad.
More generally one may ask how the lifted functorT becomes a monad with- out T being required to be a monad. Then of course some other data must be given. For an R-algebra A and an R-module V this was considered in Brzezi´nski [12, Proposition 2.1]. The transfer of this construction to endo- functors should come out as follows:
4.8. Liftings as monads. LetF= (F, µ, η) be a monad and T :A→Aany functor on the categoryAwith a natural transformation ι:I →T satisfying T ι = ιT. Then T F induces a monad (T F, µ, η) on AF with unit η = T η ◦ι and commuting diagram
T F F T µ //
T F ιF
T F T F T F
µ
::u
uu uu uu uu
if and only if there are natural transformations
λ:F T →T F and σ :T T →T F,
such that λ implies the commutative diagrams in 4.4(1), and for λ and σ there are commutative diagrams
T T T σT //
T σ
T F T T λ //T T F σF //T F F
T µ
T T F σF //T F F T µ //T F, F T T λT //
F σ
T F T T λ //T T F σF //T F F
T µ
F T F λF //T F F T µ //T F, F F ι //
ιFCCCCCC!!
CC F T
λ
T F,
T ιT //
T ηCCCCCCC!!
C T T
σ
T F.
In the terminology used in [17, Definition 1.2], this means that σ is normal (right triangle) and is a cocycle (first rectangle) satisfying the twisted module condition (second rectangle). As a special case one may take F to be the identity functor. Then the conditions reduce to T being a monad.
Given λ and σ the multiplication µis obtained by the diagram
T F T F µ //
T λF
T F T T F F σF F //T F F F T µF //T F F.
T µ
OO
If the monad (T F, µ, η) is given, suitable λ and σ are defined by the diagrams
F T λ //
ιF T
T F T F T T F T η//T F T F,
µ
OO T T σ //
T ηT
T F T F T T F T η//T F T F.
µ
OO
Dual to the constructions considered in 4.4 one obtains
4.9. Lifting of comonads to comonads. Let G = (G, δ, ε) be a comonad and T :A→A any functor on the category A.
(1) The liftings Tb : AG → AG of T are in bijective correspondence with the natural transformations ϕ : T G → GT inducing the commutative diagrams
T G T δ //
ϕ
T GG ϕG //GT G
Gϕ
GT δT //GGT,
T G T ε //
ϕ
T GT.
εT
=={
{{ {{ {{ {
(2) If T = (T, δ0, ε0) is a comonad, then the lifting Tb: AG →AG of T with natural transformation ϕ : T G → GT is a comonad if and only if we have the commutative diagrams
T G δ
0 G //
ϕ
T T G T ϕ //T GT
ϕT
GT Gδ
0 //GT T,
T G ε
0 G //
ϕ
G GT.
Gε0
==z
zz zz zz z
(3) For a comonadT= (T, δ0, ε0), a natural transformationϕ:T G→GT in- duces a canonical comonad structure onT G if and only if the diagrams in (1) and (2) are commutative.
Proof. (1) is a special case of 3.5 and the diagram shows that ϕA is a G-comodule morphism for anyA∈Obj(A).
(2) The diagrams are derived from the conditions thatδ0G(A)andε0G(A)must be G-comodule morphisms for all A ∈ Obj(A). This is seen by arguments dual to those of the proof of 4.4.
(3) This goes back to Barr [3, Theorem 2.2]. tu Similar to the composition for monads, a canonical comonad structure on T Gneed not imply that T and G are comonads.
4.10. Definition. Given two comonads G = (G, δ, ε) and T = (T, δ0, ε0) on a categoryA, a natural transformation ϕ :T G→GT is said to becomonad distributive provided the diagrams in 4.9(1) and (2) are commutative.
4.11. Tensor product of coalgebras. Given two R-coalgebras C, D, and anR-linear map
ϕ:C⊗RD→D⊗RC,
the tensor product C⊗RD can be made into a coalgebra by putting
∆ = (IC ⊗ϕ⊗ID)◦(∆C⊗∆D).
IfC andDare coassociative, the functors− ⊗RC and− ⊗RDare comonads on the category of R-modules. Then the coproduct defined on C ⊗RD is
coassociative and counital if and only if − ⊗RC ⊗RD is a comonad for the R-modules, that is, ϕ has to induce commutativity of the corresponding dia- grams in 4.9. For this special case the conditions are formulated in Caenepeel, Ion, Militaru and Zhu [18, Theorem 3.4] and also in [16, 2.14].
Similar to the case of algebras (see 4.7), for a prebraiding τ on R-Mod and R-coalgebras C, D, the natural morphism
− ⊗RτC,D :− ⊗RC⊗RD→ − ⊗RD⊗RC
is comonad distributive (the diagrams in 4.9 commute) and thus induces a coassociative coproduct on C⊗RD.
In particular the twist map tw :C⊗RD→D⊗RC satisfies the conditions imposed yielding the standard coproduct on C⊗RD.
4.12. Liftings as comonads. In 4.9(2), conditions are given for the lifting of a comonad to be a comonad. Dual to the case of monads one may ask how the lifted functorTbof a comonadG= (G, δ, ε) becomes a comonad withoutT being a comonad. This can be handled similar to the constructions considered in 4.8. In particular, based on a natural transformation ϕ : T G → GT satisfying 4.9(1), natural transformations σ : T G → T T and ε : T → I are needed satisfying appropriate conditions.
5 Combining monads and comonads
In this section we consider relationships between monads and comonads.
5.1. Lifting of monads for comonads. Let G = (G, δ, ε) be a comonad and T :A→A any functor on the category A.
(1) The liftings Tb : AG → AG of T are in bijective correspondence with the natural transformations ϕ : T G → GT inducing the commutative diagrams
T G T δ //
ϕ
T GG ϕG //GT G
Gϕ
GT δT //GGT,
T G T ε //
ϕ
T GT.
εT
=={
{{ {{ {{ {
(2) If T = (T, µ, η) is a monad, then the lifting Tb : AG → AG of T with associated natural transformationϕ:T G→GT is a monad if and only if we have the commutative diagrams
T T G µG //
T ϕ
T G
ϕ
T GT ϕT //GT T Gµ //GT,
G ηG //
GηDDDDDDD!!
D T G
ϕ
GT.
Proof. (1) follows from 3.5 and the diagrams are induced by the require- ment that theϕA are G-comodule morphisms for all A∈Obj(A).
(2) These diagrams are consequences of the condition thatµA and ηA are G-comodule morphisms but they can also be read as condition for ϕA being
aT-module morphism for any A∈Obj(A). tu
5.2. Lifting of comonads for monads. Let F = (F, µ, η) be a monad and T :A→A any functor on the category A .
(1) The liftings T : AF → AF of T are in bijective correspondence with the natural transformations λ : F T → T F inducing the commutative diagrams
F F T µT //
F λ
F T
λ
F T F λF //T F F T µ //T F,
T ηT //
T ηCCCCCCC!!
C F T
λ
T F.
(2) If T = (T, δ, ε) is a comonad, then the lifting T : AF → AF of T with associated natural transformation λ : F T → T F is a comonad if and only if we have the commutative diagrams
F T F δ //
λ
F T T λT //T F T
T λ
T F δF //T T F,
F T F ε //
λ
F T F.
εF
==z
zz zz zz z
Proof. (1) follows from 3.3 and the diagrams are induced by the require- ment that theλA are F-module morphisms for any A∈Obj(A).
(2) These diagrams are consequences of the condition that δA and εA are F-module morphisms but they can also be interpretted as the condition that λA is a T-comodule morphism for any A∈Obj(A). tu
We observe that in 5.1 and 5.2 essentially the same diagrams arise.
5.3. Mixed distributive laws. Let F = (F, µ, η) be a monad and G = (G, δ, ε) a comonad on the category A. Then a natural transformation
λ:F G→GF
is said to bemixed distributiveorentwiningprovided it induces commutative diagrams
F F G µG //
F λ
F G
λ
F GF λF //GF F Gµ //GF,
F G F δ //
λ
F GG λG //GF G
Gλ
GF δF //GGF,
G ηG //
GηCCCCCCC!!
C F G
λ
GF,
F G F ε //
λ
F GF.
εF
==z
zz zz zz z
The suggestion to consider distributive laws of mixed type goes back to Beck [6, page 133] (see Remarks 5.5). The interest in these structures is based on the following theorem which follows from 5.1 and 5.2.
5.4. Characterisation of entwinings. For a monad F = (F, µ, η) and a comonad G= (G, δ, ε) on the category A, consider the diagrams
AF G //
UF
AF UF
A G //A,
AG
Fb //
UG
AG
UG
A F //A. The following conditions are equivalent:
(a) There is an entwining natural transformation λ:F G→GF; (b) G:AF →AF is a lifting of G and has a comonad structure;
(c) Fb:AG →AG is a lifting of F and has a monad structure.
5.5. Remarks. The preceding theorem was first formulated 1973 by van Osdol in [54, Theorem IV.1]. It was extended to V-categories in Wolff [57, Theorem 2.4] and was rediscovered in 1997 by Turi and Plotkin in the context of operational semantics in [52, Theorem 7.1]. In the same year the corre- sponding notion for tensor functors was considered by Brzezi´nski and Majid who coined the nameentwining structure for a mixed distributive law for an algebraAand a coalgebraC over a commutative ringRin [14, Definition 2.1]
(see 5.8). The connection between this notions is also mentioned in Hobst and Pareigis [26].
It was observed by Takeuchi that these structures are closely related to corings (see [13, Proposition 2], [16, 32.6]). This is a special case of 5.4(b) since the coringA⊗RCis just a comonad on the category of rightA-modules.
The comultiplication is a special case of the constructions considered in the next section. Similarly, by 5.4(c), C ⊗RA can be seen as a monad on the category of right C-comodules.
5.6. Comultiplication induced by units. Let F, G be endofunctors on a category A and η : I → F a natural transformation. Then we have natural transformations
ηG :G→F G, Gη :G→GF,
and naturality of η implies commutativity of the diagrams G ηG //
ηG
F G
F ηG
F G ηF G//F F G,
G Gη //
Gη
GF
GηF
GF GF η//GF F.
For F = G the diagrams show that both ηF and F η induce coassociative comultiplications onF.
If there is a coassociative comultiplicationδ:G→GG, then we can define a comultiplication on F G by
δ¯:F G F δ //F GG F GηG//F GF G, which is coassociative by commutativity of the diagram
F G F δ //
F δ
F GG F GηG //
F Gδ
F GF G
F GF δ
F GG F δG //
F GηG
F GGG F GηGG //
F GGηG
F GF GG
F GF GηG
F GF G F δF G //F GGF G F GηGF G //F GF GF G.
The left top rectangle commutes by coassociativity ofδ, the right top rectangle by naturality ofη, the left bottom rectangle by naturality ofδ and the right bottom rectangle again by naturality of η.
For a monadF= (F, µ, η) the comultiplication onF Gcan also be derived from general properties of adjoint functors.
Symmetrically a coassociative comultiplication for GF is defined by δ˜:GF δF //GGF GηGF//GF GF.
In case a natural transformation ε : G → I is given, we have natural transformations εF : GF → F and F ε : F G → F allowing to dualise the above constructions. Then an associative multiplicationµ:F F →F induces associative multiplications onGF and F G.
Now let F= (F, µ, η) be a monad and G= (G, δ, ε) a comonad onAwith a natural transformation λ :F G→GF satisfying λ◦ηG =Gη (left triangle in 5.3). Then we have the commutative diagram
F GG
F ηLLGGLLLLLLL%%
L F GηG
**V
VV VV VV VV VV VV VV VV VV V
F G
F δuuuuuuu::
uu
F ηHHGHHHHHH$$
H F F GG F λ
G
//F GF G
F F G,
F F δ
99s
ss ss ss ss s
showing that the coproduct onF Ginduced by an entwining λis the same as the one considered above.
5.7. Mixed bimodules. Given a monad F = (F, µ, η) and a comonad G= (G, δ, ε) on the categoryAwith an entwiningλ:F G→GF,λ-bimodules ormixed bimodules are defined as those A∈Obj(A) with morphisms
F(A) h //A k //G(A)
such that (A, h) is an F-module and (A, k) is a G-comodule satisfying the pentagonal law
F(A) h //
F(k)
A k //G(A)
F G(A) λA //GF(A).
G(h)
OO
A morphismf :A→A0 between twoλ-bimodules is abimodule morphism provided it is both an F-module and aG-comodule morphism.
These notions yield the category of λ-bimodules which we denote by AGF. This category can also be considered as the category ofG-comodules for theb comonadGb:AF →AF and also as the category of F-modules for the monad F : AG →AG (e.g. [52, 7.1]). For every F-module A, G(A) is a λ-bimodule and for anyG-comoduleA0, F(A0) is aλ-bimodule canonically. In particular, for every A∈Obj(A),F G(A) and GF(A) are λ-bimodules.
As a sample we draw the diagram showing that, for any F-module %A : F(A)→ A, G(A) is a λ-bimodule with module structure given by the com- position GρA◦λA:F G(A)→G(A):
F G(A) λA //
F δA
GF(A) G%A //
δUF(A)UUUUUUUUUU**
UU UU UU
U G(A) δA //GG(A) GGF(A)
GG%A
OO
F GG(A) λG(A) //GF G(A).
GλA
OO
The triangle is commutative by naturality ofδ, the pentagon is commutative by one of the mixed distributive laws.
5.8. Entwined algebras and coalgebras. Given an R-algebra (A, µ, η) and anR-coalgebra (C,∆, ε), the functor − ⊗RA is a monad and − ⊗RC is a comonad on the category of R-modules.
